AP Calculus Review - Integration

The goal of integration by parts is to go from an integral of 𝑢 𝑑𝑣 that we don't see how to evaluate, to an integral of 𝑣 𝑑𝑢 that we can evaluate. Sometimes we have to use integration by parts more than once to evaluate an integral. And, sadly, sometimes integration by parts doesn't work. Integration by parts is derived from the product rule of differentiation.

Integration by Parts

Note the limits on the integral. Use a change in the order of the limits of an integral to have it reflect what makes sense to you.

Order of Integration

If 𝑓(𝑥) = 𝑃(𝑥)/𝑄(𝑥), where 𝑃 and 𝑄 are polynomials with the degree of 𝑃 less than the degree of 𝑄, and if 𝑄(𝑥) can be written as a product of distinct linear factors, then 𝑓(𝑥) can be written as a sum of rational functions with distinct linear denominators.

Partial Fraction Decomposition

We use a Riemann Sum to approximate the area under a curve by approximating the area with a series of rectangles. A Riemann Sum can be a left-, right-, or midpoint-rectangular approximation method. These are commonly referred to as LRAM, RRAM, or MRAM. The location (left, right or midpoint) corresponds to the top of the rectangle and the location where it intersects the given curve. The 𝑦 coordinate of each of these locations is the height of the respective rectangle for which you need to calculate the area. Typically the width of each rectangle is a uniform amount - but this is not an absolute requirement. A Riemann Sum question sometimes asks whether the sum is an overestimate or an underestimate.

Riemann Sums

A differential equation of the form 𝑑𝑦/𝑑𝑥 = 𝑓(𝑦) 𝑔(𝑥) is called separable. In this circumstance, you will want to arrange the equation with the 𝑦 variable expression and 𝑑𝑦 on one side of the = sign, and the 𝑥 variable expression and 𝑑𝑥 on the other side. Integrate both sides and solve.

Separable Differential Equation

The differential equation for this equation is: 𝑑𝑦/𝑑𝑡 = -𝑘𝑦 𝑦₀ is the initial amount. 𝑘 is the decay constant. The half life of a radioactive element is the time required for half of the radioactive nuclei present in a sample to decay.

Half Life

The differential equation for this equation is: 𝑑𝑃/𝑑𝑡 = 𝑘𝑃 (𝑀 − 𝑃 ) 𝑀 is the maximal carrying capacity. 𝐴 is the constant determined from an initial condition. 𝑘 is the growth constant. The logistic model is typically used to model population growth

Logistic Differential Equation

f(x) must be continuous on [a, b]. The point c lies on this closed interval. The quantity to the right of the = sign is referred to as the average (mean) value of the function, f(x), on the interval [a, b].

Mean Value Theorem for Definite Integrals

The rate at which an object's temperature is changing at any given time is roughly proportional to the difference between its temperature and the temperature of the surrounding medium. The differential equation for this equation is: 𝑑T/𝑑𝑡 = -𝑘 (T - Ts) 𝑇₀ is the initial amount. 𝑇s is the surrounding temperature. 𝑇 is the temperature of the object at time 𝑡.

Newton's Law of Cooling

= 0 Note the limits on the integral.

The "Zero" Integral

The differential equation for this equation is: 𝑑𝑦/𝑑𝑡 = 𝑘𝑦 𝑦₀ is the initial amount. 𝑘 is the growth/decay constant (the continuous rate of interest).

The Law of Exponential Change aka Continuous Compounded Interest

The TAM uses trapezoids instead of rectangles to approximate the area under a function. Remember that the area of a trapezoid is equal to the average base multiplied by the height. Less common, but good to know: TAM = ½ × (LRAM + RRAM) Note that the trapezoids are on their side and consecutive trapezoids share a common base.

Trapezoidal Approximation Method (TAM)